Nonlinear stability of rotating patterns and the freezing method

W.-J. Beyn, University of Bielefeld, Germany

Thursday, December 10
at 9.15, HG G 19.1

We consider nonlinear time dependent reaction diffusion systems on unbounded domains, the solutions of which show specific spatio-temporal patterns. For nonlinearities of so called excitable type, typical examples of patterns are travelling waves in one, rigidly rotating or meandering spiral waves in two, and scroll waves in three space dimensions. In the first part of the talk we present a result on stability with asymptotic phase for localized two-dimensional rotating waves (joint work with Jens Lorenz, Albuquerque). We discuss in some detail the underlying spectral assumptions for the linearized operator, which has both essential spectrum and isolated eigenvalues, some of them lying on the imaginary axis. It will be shown that the result applies to spinning solitons in the complex Ginzburg Landau equations.

In the second part we present the freezing method which transforms the given time-dependent PDE into a PDAE (Partial Differential Algebraic Equation). Solving this PDAE numerically allows to determine a moving coordinate frame in which the aforementioned patterns become stationary. The method generalizes to evolution equations that are equivariant with respect to the action of a generally noncompact Lie group. In some cases we can show that moving patterns which are stable with asymptotic phase for the original PDE become asymptotically stable in the classical Lyapunov sense for the PDAE.